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Keywords:
second-order differential equation; super-linearity; positive solution; existence; uniqueness
second-order differential equation; super-linearity; positive solution; existence; uniqueness
Summary:
We study the existence and uniqueness of a positive solution to the problem $$ u''=p(t)u+q(t,u)u+f(t);\quad u(0)=u(\omega ),\ u'(0)=u'(\omega ) $$ with a super-linear nonlinearity and a nontrivial forcing term $f$. To prove our main results, we combine maximum and anti-maximum principles together with the lower/upper functions method. We also show a possible physical motivation for the study of such a kind of periodic problems and we compare the results obtained with the facts well known for the corresponding autonomous case.
References:
[1] Cabada, A., Cid, J. Á., López-Somoza, L.: Maximum Principles for the Hill's Equation. Academic Press, London (2018). MR 3751358 | Zbl 1393.34003
[2] Chen, H., Li, Y.: Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities. Proc. Am. Math. Soc. 135 (2007), 3925-3932. DOI 10.1090/S0002-9939-07-09024-7 | MR 2341942 | Zbl 1166.34313
[3] Chen, H., Li, Y.: Bifurcation and stability of periodic solutions of Duffing equations. Nonlinearity 21 (2008), 2485-2503. DOI 10.1088/0951-7715/21/11/001 | MR 2448227 | Zbl 1159.34033
[4] Coster, C. De, Habets, P.: Two-Point Boundary Value Problems: Lower and Upper Solutions. Mathematics in Science and Engineering 205. Elsevier, Amsterdam (2006). DOI 10.1016/s0076-5392(06)x8055-4 | MR 2225284 | Zbl 1330.34009
[5] Fabry, C., Mawhin, J., Nkashama, M. N.: A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations. Bull. Lond. Math. Soc. 18 (1986), 173-180. DOI 10.1112/blms/18.2.173 | MR 0818822 | Zbl 0586.34038
[6] Fonda, A.: Playing Around Resonance: An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, Cham (2016). DOI 10.1007/978-3-319-47090-0 | MR 3585909 | Zbl 1364.34004
[7] Graef, J. R., Kong, L., Wang, H.: Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. J. Differ. Equations 245 (2008), 1185-1197. DOI 10.1016/j.jde.2008.06.012 | MR 2436827 | Zbl 1203.34028
[8] Grossinho, M. R., Sanchez, L.: A note on periodic solutions of some nonautonomous differential equations. Bull. Aust. Math. Soc. 34 (1986), 253-265. DOI 10.1017/S000497270001011X | MR 0854571 | Zbl 0592.34028
[9] Liang, S.: Exact multiplicity and stability of periodic solutions for Duffing equation with bifurcation method. Qual. Theory Dyn. Syst. 18 (2019), 477-493. DOI 10.1007/s12346-018-0296-x | MR 3982750 | Zbl 1472.34077
[10] Lomtatidze, A.: Theorems on differential inequalities and periodic boundary value problem for second-order ordinary differential equations. Mem. Differ. Equ. Math. Phys. 67 (2016), 1-129. MR 3472904 | Zbl 1352.34033
[11] Lomtatidze, A., Šremr, J.: On periodic solutions to second-order Duffing type equations. Nonlinear Anal., Real World Appl. 40 (2018), 215-242. DOI 10.1016/j.nonrwa.2017.09.001 | MR 3718982 | Zbl 1396.34024
[12] Šremr, J.: Existence and exact multiplicity of positive periodic solutions to forced non-autonomous Duffing type differential equations. Electron. J. Qual. Theory Differ. Equ. 2021 (2021), Article ID 62, 33 pages. DOI 10.14232/ejqtde.2021.1.62 | MR 4389331 | Zbl 1488.34248
[13] Šremr, J.: On a structure of the set of positive solutions to second-order equations with a super-linear non-linearity. Georgian Math. J. 29 (2022), 139-152. DOI 10.1515/gmj-2021-2117 | MR 4373259 | Zbl 1490.34049
[14] Torres, P. J.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equations 190 (2003), 643-662. DOI 10.1016/S0022-0396(02)00152-3 | MR 1970045 | Zbl 1032.34040
[15] Zamora, M.: A note on the periodic solutions of a Mathieu-Duffing type equations. Math. Nachr. 290 (2017), 1113-1118. DOI 10.1002/mana.201400122 | MR 3652218 | Zbl 1368.34039
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